Answer to Question #120975 in Mechanics | Relativity for Mamart Sanchez

Question

Calculate the moment of inertia of each of the following uniform objects about the axes indicated. A thin 2.50 kg rod lenght 75.0 cm about an axis perpendicular to it and passing a.)(i) through one end (ii) through its center, and (iii) about an axis parallel to the rod and passing through it. b.) A 3.00 kg sphere is (i) solid and (ii) A thin walled hollow shell c.) An 8.00 kg cylinder of lenght 19.5 cm cylinder if the cylinder is (i) thin-walled and hollow and (i) solid.

Answer 1

(a)

(i) $I_{end}=\frac{ml^2}{2}=\frac{2.5\cdot 0.75^2}{2}=0.703$ $kg\cdot m^2$

(ii) $I_{center}=\frac{ml^2}{12}=\frac{2.5\cdot 0.75^2}{12}=0.117$ $kg\cdot m^2$

(iii) $I_{th}=0$

(b) Assume that the radius of sphere, thin walled hollow shell, solid cylinder and thin-walled hollow cylinder is $r=1m$

(i) $I_{solid}=\frac{2}{5}mr^2=\frac{2}{5}\cdot 3\cdot 1^2=1.2$ $kg\cdot m^2$

(ii) $I_{hollow}=\frac{2}{3}mr^2=\frac{2}{3}\cdot 3\cdot 1^2=2$ $kg\cdot m^2$

(c)

(i) $I_{hollow}=mr^2=8\cdot 1^2=8$ $kg\cdot m^2$ (thin cylindrical shell with open ends)

(ii) $I_{solid, z}=\frac{1}{2}mr^2=\frac{1}{2}\cdot 8\cdot 1^2=4$ $kg\cdot m^2$

$I_{solid, x,y}=\frac{1}{12}m(3r^2+h^2)=\frac{1}{12}\cdot 8\cdot (3\cdot1^2+0.195^2)=3.04$ $kg\cdot m^2$